 Hello and how are you all today? My name is Priyanka and the question says find the mean and variance of the first n natural numbers. So here we have the first n natural numbers as 1, 2, 3, 4 and so on till n. So here the formula that we know to find out the sum of first n natural number is equal to n plus 1 upon. Now to find out the mean we need to find out the sum of all the observation that is xi from i is equal to 1 till n and we will divide it by n. So the sum is found out to be n n plus 1 upon 2 it will get divided by n. On simplification we have the answer as n plus 1 upon 2 as the mean right. So this completes the first part of the question. Now to find out the variance we need to find out first of all summation xi minus x bar the whole square. Now we know xi are all numbers that is 1, 2 and so on till n. We will be subtracting the mean that is n plus 1 upon 2 from each of them and we will find their square. So we have it equal to and using the formula a minus b the whole square we have it as 1 square plus n plus 1 upon 2 the whole square minus n plus 1. Then here we have 2 square plus n plus 1 upon 2 the whole square minus 2 n plus 1 and then till nth term we have n square plus n plus 1 upon 2 the whole square minus n into n minus 1 sorry n plus 1. Now to find out this sum we can see that if we add 1 square plus 2 square plus 3 square it can be written as n into n plus 1 into 2 n plus 1 upon 6 right and we have n times n plus 1 upon 2 the whole square right and then we can write this function as these terms on combining as minus n plus 1 which is common in each and every term and then we again have 1 2 3 and so on which can be written as n n plus 1 upon 2. So let us solve it we have n into n plus 1 into 2 n plus 1 upon 6 plus n into n plus 1 upon 2 the whole square minus n plus 1 into n n plus 1 upon 2. Now on solving this whole right inside we will get in the end n into n square minus 1 upon 12 as the sum of xi minus x bar the whole square where i is equal to 1 till n so variance is equal to summation i is equal to 1 till n xi minus x bar the whole square divided by so it is n into n square minus 1 upon 12 into n simplifying it we have the answer as n square minus 1 upon 12 right so the answer to this question is we have found out mean above as n plus 1 upon 2 remember and we have just now found the variance as n square minus 1 upon 12 right so hope you liked the session understood it too have a very nice day ahead